Fish dieing "the next day" after you buy them most likely isn't water related, in my opinion. Could be poor acclimation, or sick fish to start with. Not sure if you're using a quarantine/hospital/holding tank to isolate the new fish from your main tank, but if not... it's a good idea. That way you aren't introducing disease or parasites into your main tank. Keeping the new fish isolated for a month gives you time to recognize and treat any disease the new fella might have.
For fish only, treated tap water is *probably* just fine. "Treated" implies you're doing something to remove the chlorine or chloramines from the water. If your tank water tests zero for ammonia and nitrites, and has less than say... 40ppm nitrates... you're most likely fine.
People recommend RO
because it will not introduce any phosphates or nitrates that can fuel algae growth. That, and the fact that when you start with absolutely pure water, you have consistent salt water all the time. Consistency goes a long way to reduce stress on the fish, but more important any coral that might be in the tank.
the best you can do? Yup. But is treated tap OK for fish only? It's not the best, but unless the water is terribly terribly bad, I can't see it causing fish deaths in a day.